HSC Mathematics Advanced, Extension 1, and Extension 2/3-Unit/HSC/Primitive of sin and cos squared

The integration of these functions can be performed by the substitution of or by a function in terms of respectively.

Deriving the substituted functions


By the double-angle formula,  . Rearranging to make   the subject,  .

We know by the Pythagorean identity   that  , so  . Rearranging,  .

Finding the primitive


By substituting   for  ,  . Similarly for  ,


These can be integrated by using the primitives of cosine and sine



For cos2 α




For sin2 α